Trigonometry Examples

$y = - \frac{3}{4} x + 15$

Step 1

Interchange the variables.

$x = - \frac{3}{4} y + 15$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $- \frac{3}{4} y + 15 = x$ .

$- \frac{3}{4} y + 15 = x$

Step 2.2

Simplify each term.

Tap for more steps...

Step 2.2.1

Combine $y$ and $\frac{3}{4}$ .

$- \frac{y \cdot 3}{4} + 15 = x$

Step 2.2.2

Move $3$ to the left of $y$ .

$- \frac{3 y}{4} + 15 = x$

Step 2.3

Subtract $15$ from both sides of the equation.

$- \frac{3 y}{4} = x - 15$

Step 2.4

Multiply both sides of the equation by $- \frac{4}{3}$ .

$- \frac{4}{3} (- \frac{3 y}{4}) = - \frac{4}{3} (x - 15)$

Step 2.5

Simplify both sides of the equation.

Tap for more steps...

Step 2.5.1

Simplify the left side.

Tap for more steps...

Step 2.5.1.1

Simplify $- \frac{4}{3} (- \frac{3 y}{4})$ .

Tap for more steps...

Step 2.5.1.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.5.1.1.1.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$\frac{- 4}{3} (- \frac{3 y}{4}) = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.1.2

Move the leading negative in $- \frac{3 y}{4}$ into the numerator.

$\frac{- 4}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{3} \cdot \frac{- 3 y}{4} = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.1.1.2.1

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.3

Multiply.

Tap for more steps...

Step 2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{4}{3} (x - 15)$

Step 2.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{4}{3} (x - 15)$

Step 2.5.2

Simplify the right side.

Tap for more steps...

Step 2.5.2.1

Simplify $- \frac{4}{3} (x - 15)$ .

Tap for more steps...

Step 2.5.2.1.1

Simplify terms.

Tap for more steps...

Step 2.5.2.1.1.1

Apply the distributive property.

$y = - \frac{4}{3} x - \frac{4}{3} \cdot - 15$

Step 2.5.2.1.1.2

Combine $x$ and $\frac{4}{3}$ .

$y = - \frac{x \cdot 4}{3} - \frac{4}{3} \cdot - 15$

Step 2.5.2.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.2.1.1.3.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$y = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot - 15$

Step 2.5.2.1.1.3.2

Factor $3$ out of $- 15$ .

$y = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot (3 (- 5))$

Step 2.5.2.1.1.3.3

Cancel the common factor.

$y = - \frac{x \cdot 4}{3} + \frac{- 4}{3} \cdot (3 \cdot - 5)$

Step 2.5.2.1.1.3.4

Rewrite the expression.

$y = - \frac{x \cdot 4}{3} - 4 \cdot - 5$

Step 2.5.2.1.1.4

Multiply $- 4$ by $- 5$ .

$y = - \frac{x \cdot 4}{3} + 20$

Step 2.5.2.1.2

Move $4$ to the left of $x$ .

$y = - \frac{4 x}{3} + 20$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{4 x}{3} + 20$

Step 4

Verify if $f^{- 1} (x) = - \frac{4 x}{3} + 20$ is the inverse of $f (x) = - \frac{3}{4} x + 15$ .

Tap for more steps...

Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (- \frac{3}{4} x + 15)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 (- \frac{3}{4} x + 15)}{3} + 20$

Step 4.2.3

Simplify each term.

Tap for more steps...

Step 4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1.1

Combine $x$ and $\frac{3}{4}$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 (- \frac{x \cdot 3}{4} + 15)}{3} + 20$

Step 4.2.3.1.2

Move $3$ to the left of $x$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 (- \frac{3 \cdot x}{4} + 15)}{3} + 20$

Step 4.2.3.1.3

Factor $3$ out of $- \frac{3 x}{4} + 15$ .

Tap for more steps...

Step 4.2.3.1.3.1

Factor $3$ out of $- \frac{3 x}{4}$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 (3 (- \frac{x}{4}) + 15)}{3} + 20$

Step 4.2.3.1.3.2

Factor $3$ out of $15$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 (3 (- \frac{x}{4}) + 3 (5))}{3} + 20$

Step 4.2.3.1.3.3

Factor $3$ out of $3 (- \frac{x}{4}) + 3 (5)$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 (3 (- \frac{x}{4} + 5))}{3} + 20$

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{4 \cdot (3 (- \frac{x}{4} + 5))}{3} + 20$

Step 4.2.3.1.4

Multiply $4$ by $3$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{12 (- \frac{x}{4} + 5)}{3} + 20$

Step 4.2.3.1.5

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{12 (- \frac{x}{4} + 5 \cdot \frac{4}{4})}{3} + 20$

Step 4.2.3.1.6

Combine $5$ and $\frac{4}{4}$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{12 (- \frac{x}{4} + \frac{5 \cdot 4}{4})}{3} + 20$

Step 4.2.3.1.7

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{12 (\frac{- x + 5 \cdot 4}{4})}{3} + 20$

Step 4.2.3.1.8

Multiply $5$ by $4$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{12 (\frac{- x + 20}{4})}{3} + 20$

Step 4.2.3.2

Combine $12$ and $\frac{- x + 20}{4}$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{\frac{12 (- x + 20)}{4}}{3} + 20$

Step 4.2.3.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.3.3.1

Reduce the expression $\frac{12 (- x + 20)}{4}$ by cancelling the common factors.

Tap for more steps...

Step 4.2.3.3.1.1

Factor $4$ out of $12 (- x + 20)$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{\frac{4 (3 (- x + 20))}{4}}{3} + 20$

Step 4.2.3.3.1.2

Factor $4$ out of $4$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{\frac{4 (3 (- x + 20))}{4 (1)}}{3} + 20$

Step 4.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{\frac{4 (3 (- x + 20))}{4 \cdot 1}}{3} + 20$

Step 4.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{\frac{3 (- x + 20)}{1}}{3} + 20$

Step 4.2.3.3.2

Divide $3 (- x + 20)$ by $1$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{3 (- x + 20)}{3} + 20$

Step 4.2.3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.3.4.1

Cancel the common factor.

$f^{- 1} (- \frac{3}{4} x + 15) = - \frac{3 (- x + 20)}{3} + 20$

Step 4.2.3.4.2

Divide $- x + 20$ by $1$ .

$f^{- 1} (- \frac{3}{4} x + 15) = - (- x + 20) + 20$

Step 4.2.3.5

Apply the distributive property.

$f^{- 1} (- \frac{3}{4} x + 15) = x - 1 \cdot 20 + 20$

Step 4.2.3.6

Multiply $- - x$ .

Tap for more steps...

Step 4.2.3.6.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{3}{4} x + 15) = 1 x - 1 \cdot 20 + 20$

Step 4.2.3.6.2

Multiply $x$ by $1$ .

$f^{- 1} (- \frac{3}{4} x + 15) = x - 1 \cdot 20 + 20$

Step 4.2.3.7

Multiply $- 1$ by $20$ .

$f^{- 1} (- \frac{3}{4} x + 15) = x - 20 + 20$

Step 4.2.4

Combine the opposite terms in $x - 20 + 20$ .

Tap for more steps...

Step 4.2.4.1

Add $- 20$ and $20$ .

$f^{- 1} (- \frac{3}{4} x + 15) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (- \frac{3}{4} x + 15) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (- \frac{4 x}{3} + 20)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{4 x}{3} + 20) = - \frac{3}{4} \cdot (- \frac{4 x}{3} + 20) + 15$

Step 4.3.3

Simplify each term.

Tap for more steps...

Step 4.3.3.1

Apply the distributive property.

$f (- \frac{4 x}{3} + 20) = - \frac{3}{4} \cdot (- \frac{4 x}{3}) - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.3.2.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$f (- \frac{4 x}{3} + 20) = \frac{- 3}{4} \cdot (- \frac{4 x}{3}) - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.2.2

Move the leading negative in $- \frac{4 x}{3}$ into the numerator.

$f (- \frac{4 x}{3} + 20) = \frac{- 3}{4} \cdot \frac{- 4 x}{3} - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.2.3

Factor $3$ out of $- 3$ .

$f (- \frac{4 x}{3} + 20) = \frac{3 (- 1)}{4} \cdot \frac{- 4 x}{3} - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.2.4

Cancel the common factor.

$f (- \frac{4 x}{3} + 20) = \frac{3 \cdot - 1}{4} \cdot \frac{- 4 x}{3} - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.2.5

Rewrite the expression.

$f (- \frac{4 x}{3} + 20) = \frac{- 1}{4} \cdot (- 4 x) - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.3.3.1

Factor $4$ out of $- 4 x$ .

$f (- \frac{4 x}{3} + 20) = \frac{- 1}{4} \cdot (4 (- x)) - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.3.2

Cancel the common factor.

$f (- \frac{4 x}{3} + 20) = \frac{- 1}{4} \cdot (4 (- x)) - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.3.3

Rewrite the expression.

$f (- \frac{4 x}{3} + 20) = - 1 (- x) - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.4

Multiply $- 1$ by $- 1$ .

$f (- \frac{4 x}{3} + 20) = 1 x - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.5

Multiply $x$ by $1$ .

$f (- \frac{4 x}{3} + 20) = x - \frac{3}{4} \cdot 20 + 15$

Step 4.3.3.6

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.3.3.6.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$f (- \frac{4 x}{3} + 20) = x + \frac{- 3}{4} \cdot 20 + 15$

Step 4.3.3.6.2

Factor $4$ out of $20$ .

$f (- \frac{4 x}{3} + 20) = x + \frac{- 3}{4} \cdot (4 (5)) + 15$

Step 4.3.3.6.3

Cancel the common factor.

$f (- \frac{4 x}{3} + 20) = x + \frac{- 3}{4} \cdot (4 \cdot 5) + 15$

Step 4.3.3.6.4

Rewrite the expression.

$f (- \frac{4 x}{3} + 20) = x - 3 \cdot 5 + 15$

Step 4.3.3.7

Multiply $- 3$ by $5$ .

$f (- \frac{4 x}{3} + 20) = x - 15 + 15$

Step 4.3.4

Combine the opposite terms in $x - 15 + 15$ .

Tap for more steps...

Step 4.3.4.1

Add $- 15$ and $15$ .

$f (- \frac{4 x}{3} + 20) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (- \frac{4 x}{3} + 20) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{4 x}{3} + 20$ is the inverse of $f (x) = - \frac{3}{4} x + 15$ .

$f^{- 1} (x) = - \frac{4 x}{3} + 20$